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30MSC 2023 #10: Magic Numbers, 9 And 11.

30MSC 2023 #10: Magic Numbers, 9 and 11.

In his Vedic Mathematics Teacher Training Course, Sir Kenneth Williams discussed extensively two numbers, 9 and 11, which are on opposite sides of the base 10. He called them magic numbers probably because their varied applications not only in arithmetic but also in Algebra.

A group of problems commonly given to grade school kids involves determining the remainder when a number is divided by a certain divisor. Last Dec 4, 2023, we posted 30MSC 2023 #4: Digit Sums where we showed that the remainder when a number is divided by 9 is its digit sum.  This naturally follows the divisibility rule for 9.

Now the divisibility rule for 11 is: if the total of the digits in the odd places minus the total of the digits in the even places is a multiple of 11, then the number is divisible by 11. Thecount should begin with the units digit moving to the left.

Following this rule, we can say that the remainder when a number is divided by 11 is the difference of the total of the digits in odd places minus the total of digits in the even places.

This can be easily seen if we take some simple examples:

  1. 25 ÷ 11 = 2 r. 3; 5 – 2 = 3
  2. 331 ÷ 11 = 30 r. 1; (1 + 3) – 3 = 1
  3. 229 ÷ 11 = 20 r 9; ( 9 – 2) – 2 = 9

Thus, this question for the Junior group of the 2nd International Vedic Mathematics Olympiad(IVMO 2022) can be easily answered as (9 + 7 + 5 + 3 + 1) – (8 + 6 + 4 + 2) = 25 – 20 = 5

0ther applications of  multiplying and dividing by 9 and 11, and their extensions in polynomial operations and factoring are discussed in Chapter 10 of our e-book, 30 Master Strategies in Computing, Volume I.  The rewritten version will be available before the end of the year. Follow us on our Facebook page, MATH-Inic Philippines at https://www.facebook.com/MATHInicPhils to see tomorrow’s 30MSC 2023 #11: Using a Base

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